Grant: GA21-02411S
from 01/01/2021
to 31/12/2023
Grantor: Czech Science Foundation
Solving ill posed problems in the dynamics of compressible fluids
Objectives:
The project focuses on mathematical models of compressible fluids that are ill-posed in the framework of the existing theory. A well known example is the Euler system describing a compressible perfect gas. By solving them we mean developing suitable consistent approximation, identifying the class of limits of approximate solutions, and designing appropriate numerical methods.
Matematický ústav AV ČR usiluje o HR Award - zavedení profesionálního řízení lidských zdrojů
Matematický ústav je přední českou věřejnou organizací, kterých posláním je vědecký výzkum v oblastech matematiky a jejích aplikací. Pro posílení jeho konkurenceschopnosti v mezinárodním kontextu je klíčové uvést dosavadní strategii řízení a rozvoje lidských zdrojů do souladu s Evropskou chartou pro výzkumné pracovníky, a tím umožnit získání ocenění HR Award. Pro ústav jde o mimořádnou příležitost, jak zkvalitnit a zprofesionalizovat péči o lidské zdroje, které jsou alfou a omegou jeho úspěchu.
Tento projekt je podpořen z operačního programu Výzkum, vývoj a vzdělávání, Výzva č. 02_18_054 pro Rozvoj kapacit pro výzkum a vývoj II v prioritní ose 2 OP, reg. č. CZ.02.2.69/0.0/0.0/18_054/0014664.
Grant: GX20-31529X
from 01/01/2020
to 31/12/2024
Grantor: Czech Science Foundation
Abstract convergence schemes and their complexities
Objectives:
Abstract convergence schemes are basic category-theoretic structures which serve as universes for studying infinite evolution-like processes and their limiting behavior. Convergence schemes endowed with extra structures provide an applicable framework for studying both discrete and continuous processes as well as their random variants.
The main goal of the project is unifying and extending several concepts from model theory, algebra, topology and analysis, related to generic structures. We propose studying selected topics within the framework of abstract convergence schemes, addressing questions on their complexity and classification. One of our inspirations is the theory of universal homogeneous models, where convergence of finite structures is involved. Another motivation is set-theoretic forcing, where a convergence scheme is simply a partially ordered set of approximations of some ``unreachable" objects, living outside of the universe of set theory.
Grant: GJ20-17488Y
from 01/01/2020
to 31/12/2022
Grantor: Czech Science Foundation
Applications of C*-algebra classification: dynamics, geometry, and their quantum analogues
Objectives:
The 3 year project will bring together 4 promising mathematicians to build upon the successes of the classification programme for C*-algebras. There are 3 main goals:
A-Study structural properties and establish classification theorems for topological dynamical systems and their C*-algebras
B-Examine quantum group actions and the structure of their associated homogeneous spaces
C-Apply metric and geometric structures in C*-algebras to understand the fine structure of classifiable C*-algebras
Each goal consists of 3-4 concrete objectives. The expected output of 12-15 high quality research papers is outlined in an achievable schedule, divided into 4 month blocks. Already home to several researchers in C*-algebras, the Project's successful funding would position the Host Institute as a major European centre for C*-algebras. The team is highly active in the C*- algebra community and collaborates with world-leading experts, promising a successful outcome and efficient dissemination of results.
Symmetries, dualities and approximations in derived algebraic geometry and representation theory
Objectives:
The project focuses on new trends in homological algebra, represenation theory and algebraic geometry. In particular, we aim at studying and developing a theory of the exotic versions of derived categories and equivalences of these, studying derived commutative algebra, algebraic geometry and representation theory, and studying the homological algebra of and the structure theory for contramodules over topological rings, which were discovered only a few year ago. The applicants with collaborators recently published their results in distinguished mathematical journals (J. reine angew. Math., Invent. Math., Adv. Math., Mem. Amer. Math. Soc. and others), and the proposed project naturally builds on these results.
Grant: GA20-01074S
from 01/01/2020
to 31/12/2022
Grantor: Czech Science Foundation
Adaptive methods for the numerical solution of partial differential equations: analysis, error estimates and iterative solvers
Objectives:
The project deals with the numerical solution of several types of partial differential equations (PDEs) describing various practical phenomena and problems. The aim is to develop reliable and efficient numerical methods allowing to obtain approximate solutions of PDEs under the given tolerance using a minimal number of arithmetic operations. The whole process includes the proposals and analysis of discretization schemes together with suitable solvers for the solution of arising algebraic systems, a posteriori error estimation including algebraic errors and adaptive techniques balancing various error contributions. We focus on the use of adaptive higher-order schemes which allow to reduce significantly the number of necessary degrees of freedom required for the achievement of the prescribed accuracy. The adaptive mesh refinement must also take into account the properties of the resulting algebraic systems. The expected outputs of this projects are adaptive reliable and efficient numerical methods for the solution of the considered types of PDEs.
Grant: GA20-14736S
from 01/01/2020
to 31/12/2022
Grantor: Czech Science Foundation
Hysteresis modeling in mathematical engineering
Objectives:
Rate-independent hysteresis memory is known to occur in many physical processes such as magnetization of ferromagnetic materials, fluid flow through porous media, and phase transitions. Theoretical understanding of hysteresis mechanisms is of a key importance in engineering applications, where neglecting dissipative hysteresis effects in numerical predictions may lead to error accumulation and discrepancies with experiments. Most of the modern multifunctional materials used for high-precision devices exhibit hysteresis, which has to be taken into account in mathematical modeling. Surprisingly, hysteresis is also present in economic models. We focus here on mathematical and computational aspects of hysteresis in the whole range of applications. Special attention will be paid to the theoretical analysis of typical problems arising in dealing with smart materials, water-ice phase transitions in porous solids, and economics.
Operads are objects formalizing compositionsof operations with several inputs. They were invented to describe homotopy invariant structures on topological spaces. Later it turned out that they can be used as well for the study of sundry structures in geometry, algebra and mathematical physics.
The research supported by Praemium Academie is aimed at formulating a unifying paradigm for very general operadic structures, and using this emerging systematic approach for proving various results in algebra, geometry and mathematical physics. Our team is international from the very beginning, as emphasized by the planned positions for postdocs and foreign specialists.
Grant: GX19-27871X
from 01/01/2019
to 31/12/2023
Grantor: Czech Science Foundation
Efficient approximation algorithms and circuit complexity
Objectives:
The goal of this project is to understand the role of approximation in fine-grained and parameterized complexity and create solid foundations for these areas by developing lower bound techniques capable of addressing the key unproven assumptions under-pinning these areas. We will focus on several central problems: Edit Distance, Integer Programming, Satisfiability and study their approximation and parameterized algorithms with the aim of
designing the best possible algorithms. Additionally we will focus on several methods of proving complexity lower bounds.
Grant: GA19-09659S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Exact solutions of gravity theories: black holes, radiative spacetimes and electromagnetic fields
Objectives:
Exact solutions to Einstein gravity play a crucial role in the understanding of many mathematical and physical aspects of the theory. In recent years, for several theoretical reasons, various modifications of Einstein gravity and their solutions have been studied. Due to the complexity of resulting field equations, very few exact solutions of such theories are known. We plan to construct and study exact solutions to Einstein gravity and various higher-order
gravities, such as quadratic gravity and Lovelock gravity, with a strong focus on black hole solutions, radiative spacetimes, and p-form fields. We also intend to study generic properties of certain classes of spacetimes, such as asymptotically flat spacetimes. When appropriate, we will benefit from employing mathematical methods, such as algebraic classification and a generalized GHP formalism developed in part by our team.
Grant: GA19-04243S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Partial differential equations in mechanics and thermodynamics of fluids
Objectives:
Partial differential equations in mechanics and thermodynamics of fluids are a mathematical tool which models the time evolution of basic physical quantities. The goal of this project is to study these systems of partial differential equations from the point of view of mathematical and numerical analysis and to compare these results with the properties of their numerical solutions. We will mostly deal with solvability of the problems (existence of solutions for different formulations, possibly their uniqueness), qualitative properties of the solutions, analysis of the adequate numerical methods and numerical solutions of these problems. The proposal of the project assumes a close collaboration of specialists from different branches. Such a collaboration stimulates positively the developement of all participating mathematical disciplines.
Grant: GA19-05497S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Complexity of mathematical proofs and structures
Objectives:
We study weak logical systems, guided by the question: what is the weakest natural theory in which we can prove a mathematical statement? This question is often fundamentally complexity theoretic in nature, as proofs in such weak systems can be associated with feasible computations. We will study this and related topics in a range of settings, including bounded arithmetic, model theory, algebraic complexity, bounded set theory, and nonclassical logics.
Grant: GJ19-07129Y
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Linear-analysis techniques in operator algebras and vice versa
Objectives:
The theory of normed spaces and their operators is at the core of the linear analysis. The idea of employing algebras of operators acting on infinite-dimensional spaces originated in quantum physics and was further successfully integrated with the theory of unitary representations of locally compact groups. An operator algebra, which is also a normed space, carries intrinsically a much richer structure and therefore operator algebras are not usually viewed from the perspective of linear analysis. Nevertheless, the transfer of ideas from Banach spaces can be very fruitful as illustrated by the notion of nuclearity that was recognised as an approximation property with respect to a certain class of finite-rank operators. On the other hand, operator Ktheory
was almost unknown in Banach space theory until it was spectacularly applied in the seminal work of Gowers and Maurey. Consequently, there is tremendous potential in transferring ideas between these two areas. The very aim of the project is a closer reconciliation of these two theories by interchanging ideas between them.
Grant: GJ19-05271Y
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Groups and their actions, operator algebras, and descriptive set theory
Objectives:
The goal of the project is to to find new connections and prove some interesting conjectures on the boundaries of three, currently very attractive mathematical disciplines - geometric group theory, operator algebras, and descriptive set theory.